Hecke algebras, Specht modules and Gröbner–Shirshov bases

Abstract

In this paper, we study the structure of Specht modules over Hecke algebras using the Gröbner–Shirshov basis theory for the representations of associative algebras. The Gröbner–Shirshov basis theory enables us to construct Specht modules in terms of generators and relations. Given a Specht module S q λ , we determine the Gröbner–Shirshov pair ( R q , R q λ ) and the monomial basis G ( λ ) consisting of standard monomials. We show that the monomials in G ( λ ) can be parameterized by the cozy tableaux. Using the division algorithm together with the monomial basis G ( λ ), we obtain a recursive algorithm of computing the Gram matrices. We discuss its applications to several interesting examples including Temperley–Lieb algebras.